Galois theory allows one to reduce certain problems in field theory, especially those related to field extensions, to problems in group theory. For questions about field theory and not Galois theory, use the (field-theory) tag instead. For questions about abstractions of Galois theory, use ...

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Splitting field of polynomial over $\mathbb{F}_3$

Let $K$ be the splitting field of the polynomial $(x^3 + x - 1)(x^4 + x - 1)$ over $\mathbb{F}_3$. How many elements does $K$ contain? What I've already done is factoring $(x^3 + x - 1)(x^4 + x - 1)$ ...
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70 views

Vector Space isomorphisms of $\mathbb{Q}(z)$ preserving the Galois group (where $z$ is a primitive third root of unity)

Take the field extension $\mathbb{Q}(z)$ where $z$ is a primitive third root of unity and consider the set $A$ of vector-space automorphisms of $\mathbb{Q}(z)$ so that for $T \in A$ the map $\phi ...
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1answer
444 views

Patterns in $GF(2)$ Polynomial division.

I am testing Prime polynomials in $GF(2)$ and have noticed a pattern that I hope will help. There's a calculator here if you want to familiarise yourself with polynomials over $GF(2)$. I am testing ...
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Degree of factor in a resolvent

Background and lemma first: Let $\Theta \in k[x_1,...,x_n]^H$ and $\theta$ be its evaluation to the roots of a fixed $n$:th degree polynomial in $k[x]$. Put $L(t) = \prod_{\sigma \in S_n//H} (t-\sigma ...
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Group action on an ultraproduct

Let $\mathcal U$ be an ultrafilter, say on $\mathbb N$, and suppose we have a set $X$ with a group action by a group $G$ (for instance, one can think of $X$ as an algebraically closed field and $G$ as ...
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1answer
80 views

Normal extension

Can anybody help me with this problem ? Let $P\in {\mathbb Q}[X]$ be an irreducible polynomial of degree $p$ prime and let $z_1, z_2, \ldots ,z_p\in {\mathbb C}$ be the roots of $P$. Suppose that ...
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1answer
51 views

What could be said about a field extension, if it's Galois Group is fixpointfree?

Let $L/K$ be a galois extension, and let the Galoisgroup be fixpoint-free, what could be said about the field extension.
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106 views

Cyclotomic Polynomials over $\mathbb Q$ and reduction modulo $p$.

Let $p$ be prime, and let $\pi : \mathbb Z \to \mathbb Z / (p\mathbb Z)$ be the canonical projection $\pi(z) = z + p\mathbb Z$. Define its extension $\pi : \mathbb Z[x] \to \mathbb Z/(p\mathbb Z)[x]$ ...
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161 views

Question about the Galois extension of a given field extension

Let $K=\mathbb{Q}(\omega)$ be given, where $\omega^3=1$. I want to know: (1) Whether there is a Galois extension $L/\mathbb{Q}$ containing $K$ such that $\mathrm{Gal}(L/\mathbb{Q})\cong\mathbb{Z}_4$? ...
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1answer
76 views

What's wrong here when I compute $\operatorname{Gal} (x^4 -2 / \mathbb{Q})$

Maybe that's a stupid question and I'm missing something very trivial. Let $f(x) = x^4 - 2$ and $\alpha_1 = \sqrt[4]2, \alpha_2 = -\sqrt[4]2, \alpha_3 = i\sqrt[4]2, \alpha_4 = -i\sqrt[4]2$ be its ...
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245 views

Solvability by radicals of an equation of prime degree

For which prime $p$ the equation $x^p-p^px+p=0$ is solvable by radicals? I don't know how to solve this for primes $p\neq 2$, so any help is welcome.
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1answer
42 views

Where does the minimal polynomial lie?

Let $\theta = a + b \sqrt{D_1} + c \sqrt{D_2} + d \sqrt{D_1 D_2}$, where $a,b,c,d,D_1,D_2$ are integers. Is there any reason to believe the minimal polynomial for $\theta$ over $\mathbb{Q}$ should be ...
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2answers
425 views

Proof of Galois' theorem that there exists a field of $p^n$ elements.

From Galois Theory (Rotman): For every prime p and every positive integer n, there exists a field having exactly $p^n$ elements. Proof. If there were a field K with $|K| = p^n = q$, then ...
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1answer
146 views

Lower Bound on Number of Wildly Ramified Extensions of $\mathbb{Q}_p$

Suppose $p$ is prime and $p$ divides $e$ divides $n$. Typically up to isomorphism there are a lot of wildly ramified extensions of $\mathbb{Q}_p$ which have degree $n$ and ramification index $e$. ...
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1answer
67 views

When a subfield of a splitting field is a splitting field

Look at the following proposition: Let $K\subset L\subset M$ be three fields. If $M$ is a splitting field over $K$ of a polinomial in $K[X]$ and moreover if for every $\sigma\in G=Gal(M/K)$ we ...
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2answers
41 views

Galois extension corresponding to $S_n$

Given a positive integer $n$. How to find a Galois extension $K/F$ such that $Gal(K/F)=S_n$? With the restriction $F=\mathbb{Q}$, this is the Inverse Galois Problem. But if we are allowed to choose ...
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1answer
74 views

Galois group of a reducible polynomial over a arbitrary field

How to proceed to determine the Galois group of a reducible polynomial over a field $F$. As an example I tried to compute the Galois group of $f(X)=X^4+4\in\mathbb{Q}[X]$; one can check that ...
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1answer
48 views

Finding $\operatorname{gal}(f\mid\Bbb C(\omega))$ and $\operatorname{gal}(f\mid\Bbb R(\omega))$

How to solve following: Let $\Bbb C(\omega)$ be a field of rational functions with one undetermined $\omega$ over the field of complex numbers $\Bbb C$. If $f(x)=x^5+\omega$, find a) ...
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1answer
114 views

Galois group of $\mathbb{Q}(\sqrt[3]{2},\omega,\sqrt{-1})$ over $\mathbb{Q}$

How to find the Galois group $Gal(E|Q)$, where $E=\mathbb{Q}(\sqrt[3]{2},\omega,\sqrt{-1})$? I know that $\text{Gal}(\mathbb{Q}(\sqrt[3]{2},\omega)|\mathbb{Q})=D_3$, where $D_3$ is the dihedral group, ...
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Gaussian periods

Let p be an odd prime number and $p-1=m d$ a decomposition into positive factors. Then there is a unique cyclic extension $K_d/\mathbb Q$ of degree d ramified only at p. It is contained in the ...
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1answer
45 views

Confirm the meaning of Prime and Primitive in a Galois(2) polynomial.

Here it discusses primality (or more accurately irreducibility) and primitivity of ...
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68 views

If a monic $f\in\overline{K}[x]$ has a power $f^n\in K[x]$, where the characteristic of $K$ doesn't divide $n$, then must $f\in K[x]$?

Suppose you have a monic polynomial $f(x) \in \overline{K}[x]$, and some integer $n>1$, where $\mathrm{char}(K)\nmid n$, and $\big (f(x)\big )^n\in K[x]$. Does it imply $f(x) \in K[x]$? The ...
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176 views

Relationship between conjugate intermediate fields and conjugate subgroups of the Galois group.

Let $L/K$ be a field extension, and let $K \subseteq A \subseteq L$ an intermediate field. Define $$ A^{FG} := \operatorname{Gal}(L/A) = \{ \phi \in \operatorname{Aut}(L) : \forall x \in A: \phi(x) ...
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2answers
85 views

Isomorphic multiplicative groups of quadratic extensions

What are all ordered pairs $(n,m)$ such that the multiplicative groups of the fields $\mathbb{Q}(\sqrt{n})$ and $\mathbb{Q}(\sqrt{m})$ are isomorphic? I saw a question earlier today claiming that ...
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2k views

Self teaching Galois Theory

At uni, I did a module in group theory which I really enjoyed. I also did one on number theory which had aspects of rings and fields in it and I enjoyed learning this. Now that I have finished uni, I ...
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Abelian extensions with squarefree discriminant

Is it true that for all $2 \leq n \in \mathbb N$ we can find an abelian extension of the rationals with squarefree discriminant and degree n? Are there even infinitely many? Some even degrees may be ...
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1answer
103 views

Question on Galois extension of field of fractions

I would like to ask if the following is true: Suppose we have an integral domain $A$ and a group action $G$ on $A$. We consider $A^{G}$ the subring of $A$ fixed by $G$. Let $L$ be the field of ...
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1answer
59 views

$a\in \mathbb{C}$ for which $[ \mathbb{Q}(a) : \mathbb{Q}(a^3) ] = 2$.

I want to find a $a\in \mathbb{C}$ for which $[ \mathbb{Q}(a) : \mathbb{Q}(a^3) ] = 2$. Any ideas? Thanks.
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1answer
114 views

Discriminant of Finite-Dimensional Extension of $\mathbb{Q}_p$

For an $n$-dimensional extension $K$ of $\mathbb{Q}_p$, we have $K$'s "ring of integers" $\mathcal O_K$ and its uniformizer $\varpi$. We also have the ring of $p$-adic integers $\mathbb{Z}_p$, with ...
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1answer
60 views

How to convert a polynomial in $GF(p^n)$ into the form $a(x)^k$, with $a(x)$ a generator polynomial.

I basically have two questions: Given $GF(p^n$) and $g(x)$ an irreducible polynomial: My textbook says that a polynomial is called primitive if $x$ is a generator of the field. The question now is, ...
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1answer
435 views

Radical extensions

Suppose we have $$ K \subset K(a_1) \subset K(a_1, a_2) \subset K(a_1, a_2, a_3) $$ such that $a_j^{p_j}\in K_{j-1}$: a radical extension in other words. I am having trouble understanding why for ...
3
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1answer
218 views

Galois group of $K(X)/K$

Let $K$ be an infinite field, if $K(X)$ is the field of rational function I want to find the Galois group of the extension $K(X)/K$. Lemma 1: If $L$ is a field such that $K\subsetneq L\subseteq ...
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1answer
106 views

Finite fields as splitting fields

hey guys so i stumbled upon an example that begins with "Consider GF(25). This can be constructed as the splitting field of $t^2 - 2...$" But the theorem states that it is the splitting field of the ...
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1answer
92 views

Understanding Galois groups, too simple argumentation?

I am currently studying Galois Theory, and there I read the paper Analyzing the Galois Groups of Fifth-Degree and Fourth-Degree Polynomials In it the author states (page 26, case 1) "Let's look at ...
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3answers
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Determine the Galois Group of $(x^2-2)(x^2-3)(x^2-5)$

Determine all the subfields of the splitting fields of this polynomial. I chose this problem because I think to complete it in great detail will be a great study tool for all of the last chapter, as ...
4
votes
1answer
193 views

Degree of splitting field extensions

The problem states: Let $f (x) = x^3+px+q$ be an irreducible cubic polynomial with rational coefficients and let $K$ be the splitting field of $ f(x) $ over $\mathbb{Q}$. Prove that $ [K : ...
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Ramification group fixing an unramified extension

For a Galois extension of local fields $L/K$ with Galois group $G$, define the ramification groups $G_i$ by $$G_i = \{ \sigma \in G : \nu_{L}(\sigma(x) - x) \geq i+1 \text{ } \forall x \in ...
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2answers
535 views

Invariant fields of the Galois group of $x^4 + 1$

Let $f(x) = x^4 + 1 \in \mathbb{Q}[x]$. We can show that if $\alpha$ is a zero of $f(x)$, then the full set of zeros is given by $\{\alpha, -\alpha, i\alpha, -i\alpha\}$. Since $\alpha^2 = \pm i$ we ...
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1answer
450 views

splitting field of $(x^3-2)(x^3-3)$ over $\mathbb Q$

Question: What is the Galois group of $f(x)=(x^3-2)(x^3-3)$ over $\mathbb Q$, and what are the subfields which contain $\mathbb Q(\zeta_3)$? The roots of $f(x)$ are ...
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For each normal extension of a field, whose Galois group is commutative, each intermediate extension is also normal.

Let $K$ be a normal extension of the field $F$, and let the Galois group $G(K,F)$ be an Abelian group. Prove that each intermediate extension $E$ is also a normal extension. EDIT: All fields here are ...
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109 views

Each finite extension of a field, has a finite number of intermediate extensions.

Prove that every finite extension $K$ of a field $F$, has a finite number of intermediate extensions. EDIT: All fields here are of characteristic $0$, otherwise we would need to require the extension ...
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1answer
111 views

Is $C_2$ the correct Galois Group of $f(x)= x^3+x^2+x+1$?

Let $\operatorname{f} \in \mathbb{Q}[x]$ where $\operatorname{f}(x) = x^3+x^2+x+1$. This is, of course, a cyclotomic polynomial. The roots are the fourth roots of unity, except $1$ itself. I get ...
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1answer
69 views

Difference between permutation groups and galois groups?

A124938 Number of non-solvable transitive Galois groups for polynomials of degree n. +40 1 0, 0, 0, 0, 2, 4, 3, 5, 4, 15, 4, 35, 3, 27, 40, 49, 5, 91, 2, 358 (list; graph; refs; listen; ...
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1answer
88 views

unknown cases of the inverse galois problem

The problem : Given a subgroup $G$ of $S(n)$, is there a polynomial having galois group $G$ ? I found some useful theorems, and I believe to remember that upto degree $11$, all ...
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3answers
126 views

Galois group $\operatorname{Gal}(f/\mathbb{Q})$ of the polynomial $f(x)=(x^2+3)(x^2-2)$

Find the Galois group $\operatorname{Gal}(f/\mathbb{Q})$ of the polynomial $f(x)=(x^2+3)(x^2-2)$. Any explanations during the demonstration, will be appreciated. Thanks!
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1answer
91 views

special polynomial

Is there a polynomial with coefficients -1,0,1 which is irreducible over the rationals and has the alternating group as its galois group ? More concrete : Is there a polynomial of degree n, all ...
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1answer
80 views

probability of symmetric group

It is known that a random polynomial with integer coefficients, which is irreducible over the rationals has the full symmetric group as its galoisgroup with a "high" probability. I would like to ...
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2answers
181 views

Local fields and infinite extensions, basic questions

Notation throughout: Let $K$ be a discrete valuation field and $L/K$ an infinite (not necessarily Galois) extension of $K$. 1) How can/does one define a ramification index $e(L/K)$ for $L/K$? It ...
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487 views

Maximal ideals in polynomial rings over a field

Let $K$ be an algebraically closed field and let $k$ be a subfield of $K$ such that the field extension $K \mid k$ is algebraic. Let $B$ be the polynomial ring $K [x_1, \ldots, x_n]$ and let $A$ be ...
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221 views

finitely generated subfield of algebraic closure of the finite field with $p$ elements

Let $\mathbb{F}^{\operatorname{alg}}_p$ be the algebraic closure of the finite field with $p$ elements. I know that any finitely generated subfield of $\mathbb{F}^{\operatorname{alg}}_p $ is ...